The position of the sun in the sky is crucial for many applications, including solar energy generation, architecture, and agriculture. The solar angle calculator helps determine two key metrics: the Solar Altitude Angle and the Solar Azimuth Angle. These angles define the sun's precise location relative to an observer on Earth.
Solar Altitude Angle
The Solar Altitude Angle (also known as elevation angle) is the angle between the horizon and the center of the sun's disk. It ranges from 0° (when the sun is on the horizon) to 90° (when the sun is directly overhead at the zenith). A higher altitude angle means the sun's rays are more direct, leading to greater solar energy intensity.
Solar Azimuth Angle
The Solar Azimuth Angle is the horizontal angle of the sun measured clockwise from true North. Typically, North is defined as 0°, East as 90°, South as 180°, and West as 270°. This angle helps determine the direction from which sunlight is coming. For example, in the Northern Hemisphere, the sun is generally in the southern part of the sky during the day.
How the Calculation Works
Calculating these angles involves several steps and astronomical formulas. The core idea is to determine the sun's position in the sky at a specific time and location.
1. Day Number (N): This is the number of days passed since January 1st of the year. For example, January 1st is day 1, February 1st is day 32, etc.
2. Time in Solar Mean Anomaly (M): This represents the sun's position in its elliptical orbit.
M = (0.9856 * N) - 3.289 (degrees)
3. Apparent Solar Constant (A): This accounts for the slight variation in the Earth's orbital speed.
A = 23.44 * sin(M_rad) (degrees)
where M_rad is M converted to radians.
4. Mean Longitude of the Sun (L): The sun's average position in its orbit.
L = 6.2400 + (1.0000001831 * M) (radians)
This needs to be normalized to be between 0 and 2*PI.
5. Ecliptic Longitude (lambda): The actual angle of the sun along the ecliptic plane.
lambda = L + 0.0334 * sin(M_rad) + 0.00034 * sin(2*M_rad) (radians)
Normalized to be between 0 and 2*PI.
6. Declination Angle (delta): The angle between the sun's rays and the plane of the Earth's equator. This is crucial for altitude calculations.
delta_rad = asin(sin(epsilon_rad) * sin(lambda_rad))
where epsilon_rad is the obliquity of the ecliptic (approx 23.44 degrees or 0.409 radians).
delta (degrees) = asin(sin(epsilon) * sin(lambda))
7. Hour Angle (H): The angle representing the time of day relative to solar noon.
H = 15 * (time_of_day - 12 - timezone_offset_hours + equation_of_time_hours) (degrees)
The equation_of_time_hours is a correction for the difference between clock time and solar time, which can be approximated or calculated more precisely. For simplicity here, we'll use a basic formula derived from solar mean anomaly and longitude.
8. Solar Altitude Angle (alpha):alpha_rad = asin(sin(latitude_rad) * sin(delta_rad) + cos(latitude_rad) * cos(delta_rad) * cos(H_rad))alpha (degrees) = asin(sin(latitude) * sin(delta) + cos(latitude) * cos(delta) * cos(H))
9. Solar Azimuth Angle (gamma):gamma_rad = acos((sin(delta_rad) * cos(latitude_rad) - cos(delta_rad) * sin(latitude_rad) * cos(H_rad)) / cos(alpha_rad))
This angle needs careful interpretation: if cos(H) is negative (afternoon), the angle should be adjusted. A common approach is:
If sin(H) > 0, gamma is calculated directly.
If sin(H) < 0, gamma = 360 - gamma.
gamma (degrees) = acos((sin(delta) * cos(latitude) - cos(delta) * sin(latitude) * cos(H)) / cos(alpha))
This is often adjusted to be measured from North. A more robust formula for azimuth (where North is 0, East is 90) is:
atan2(sin(H_rad), cos(H_rad)*sin(latitude_rad) - tan(delta_rad)*cos(latitude_rad))
*Note: The formulas above provide a simplified overview. Precise calculations often involve lookup tables or more complex astronomical algorithms for accuracy, especially the Equation of Time and correcting for longitude and timezone. This calculator uses a common approximation.*
Use Cases
Solar Panel Installation: Determine optimal panel tilt and orientation for maximum energy capture.
Architecture and Building Design: Design buildings to utilize passive solar heating or minimize summer heat gain.
Agriculture: Understand sunlight exposure for crops.
Astronomy and Navigation: Basic celestial positioning calculations.
function degreesToRadians(degrees) {
return degrees * Math.PI / 180;
}
function radiansToDegrees(radians) {
return radians * 180 / Math.PI;
}
function calculateSolarAngles() {
var lat = parseFloat(document.getElementById("latitude").value);
var lon = parseFloat(document.getElementById("longitude").value);
var dayOfYear = parseInt(document.getElementById("dayOfYear").value);
var timeOfDay = parseFloat(document.getElementById("timeOfDay").value);
var tzOffset = parseFloat(document.getElementById("timezoneOffset").value);
var altitudeResultElement = document.getElementById("altitudeResult");
var azimuthResultElement = document.getElementById("azimuthResult");
// Basic validation
if (isNaN(lat) || isNaN(lon) || isNaN(dayOfYear) || isNaN(timeOfDay) || isNaN(tzOffset) ||
lat 90 || lon 180 || dayOfYear 366 ||
timeOfDay 24 || tzOffset 14) {
altitudeResultElement.textContent = "Invalid input";
azimuthResultElement.textContent = "Invalid input";
return;
}
// — Calculations —
// 1. Decimal Day of Year (ensure it's within range for leap year calculation if needed, though this is a simple approximation)
var N = dayOfYear;
// 2. Solar Mean Anomaly (M) in degrees
var M_deg = (0.9856 * N) – 3.289;
// 3. Eccentricity of Earth's orbit and related terms (simplified)
// We'll directly calculate Solar Transit and Declination from M and L
// 4. Mean Longitude of the Sun (L) in degrees
var L_deg = M_deg + 1.916 * Math.sin(degreesToRadians(M_deg)) + 0.020 * Math.sin(degreesToRadians(2 * M_deg)) + 282.634;
L_deg = L_deg % 360; // Normalize to 0-360
if (L_deg < 0) L_deg += 360;
// 5. Ecliptic Longitude (lambda) – For simplicity, we often use L directly or slightly adjust
var lambda_deg = L_deg; // Simplified: using L directly
// 6. Declination Angle (delta) in degrees
var epsilon = 23.439; // Obliquity of the ecliptic in degrees
var delta_rad = Math.asin(Math.sin(degreesToRadians(epsilon)) * Math.sin(degreesToRadians(lambda_deg)));
var delta_deg = radiansToDegrees(delta_rad);
// 7. Equation of Time (EoT) – simplified approximation based on M
// A more accurate EoT calculation is complex. This is a common approximation for demonstration.
var EoT_minutes = 4 * (L_deg – M_deg – 1.916 * Math.sin(degreesToRadians(M_deg)) – 0.020 * Math.sin(degreesToRadians(2 * M_deg)) – 282.634);
var EoT_hours = EoT_minutes / 60.0;
// 8. Hour Angle (H) in degrees
// Local Solar Time = Clock Time – Longitude/15 + TimeZoneOffset + EoT
// Local Apparent Solar Time (LAST) – Noon is 12:00
// Time in hours from midnight
var localSolarTime = timeOfDay – (lon / 15.0) + tzOffset + EoT_hours;
// Hour angle is the time difference from solar noon (12:00 LAST)
var H_deg = 15.0 * (localSolarTime – 12.0);
// — Altitude Angle Calculation —
var lat_rad = degreesToRadians(lat);
var delta_rad_calc = degreesToRadians(delta_deg); // Use the calculated declination
var H_rad = degreesToRadians(H_deg);
var sin_altitude = Math.sin(lat_rad) * Math.sin(delta_rad_calc) + Math.cos(lat_rad) * Math.cos(delta_rad_calc) * Math.cos(H_rad);
var altitude_deg = radiansToDegrees(Math.asin(sin_altitude));
// — Azimuth Angle Calculation —
// Using atan2 for robust azimuth calculation (North = 0, East = 90, South = 180, West = 270)
var denominator = Math.cos(lat_rad) * Math.cos(delta_rad_calc) * Math.sin(H_rad);
var numerator = Math.cos(delta_rad_calc) * Math.sin(lat_rad) * Math.cos(H_rad) – Math.sin(delta_rad_calc) * Math.cos(lat_rad);
var azimuth_rad = Math.atan2(denominator, numerator);
var azimuth_deg = radiansToDegrees(azimuth_rad);
// Normalize azimuth to 0-360 degrees (where 0 is North, positive is East)
// Azimuth calculated by atan2 is often -180 to 180 or similar. Let's adjust.
// If the standard formula yields a negative value, add 360.
azimuth_deg = (azimuth_deg + 360) % 360; // Ensure positive angle 0-360
// More precise Azimuth: using atan2(sin(H), cos(H)*sin(lat) – tan(delta)*cos(lat))
var atan2_num = Math.sin(H_rad);
var atan2_den = Math.cos(H_rad) * Math.sin(lat_rad) – Math.tan(delta_rad_calc) * Math.cos(lat_rad);
var azimuth_rad_v2 = Math.atan2(atan2_num, atan2_den);
var azimuth_deg_v2 = radiansToDegrees(azimuth_rad_v2);
// Normalize v2 to 0-360 where North is 0
azimuth_deg_v2 = (azimuth_deg_v2 + 360) % 360;
// Handle cases where the sun is below the horizon
if (altitude_deg < 0) {
altitude_deg = 0; // Or indicate it's below horizon
azimuthResultElement.textContent = "– (Sun below horizon)";
} else {
azimuthResultElement.textContent = azimuth_deg_v2.toFixed(2);
}
altitudeResultElement.textContent = altitude_deg.toFixed(2);
}